Properties

Label 2-300-300.287-c2-0-66
Degree $2$
Conductor $300$
Sign $0.977 - 0.212i$
Analytic cond. $8.17440$
Root an. cond. $2.85909$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.82 − 0.820i)2-s + (1.40 + 2.65i)3-s + (2.65 − 2.99i)4-s + (−3.62 + 3.44i)5-s + (4.73 + 3.67i)6-s + (8.40 − 8.40i)7-s + (2.38 − 7.63i)8-s + (−5.04 + 7.45i)9-s + (−3.78 + 9.25i)10-s + (15.5 + 11.3i)11-s + (11.6 + 2.82i)12-s + (−1.80 + 11.3i)13-s + (8.43 − 22.2i)14-s + (−14.2 − 4.76i)15-s + (−1.92 − 15.8i)16-s + (−5.37 − 10.5i)17-s + ⋯
L(s)  = 1  + (0.911 − 0.410i)2-s + (0.468 + 0.883i)3-s + (0.663 − 0.748i)4-s + (−0.725 + 0.688i)5-s + (0.789 + 0.613i)6-s + (1.20 − 1.20i)7-s + (0.297 − 0.954i)8-s + (−0.560 + 0.827i)9-s + (−0.378 + 0.925i)10-s + (1.41 + 1.02i)11-s + (0.971 + 0.235i)12-s + (−0.138 + 0.876i)13-s + (0.602 − 1.58i)14-s + (−0.948 − 0.317i)15-s + (−0.120 − 0.992i)16-s + (−0.316 − 0.620i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.212i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.977 - 0.212i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ 0.977 - 0.212i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.28612 + 0.352451i\)
\(L(\frac12)\) \(\approx\) \(3.28612 + 0.352451i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.82 + 0.820i)T \)
3 \( 1 + (-1.40 - 2.65i)T \)
5 \( 1 + (3.62 - 3.44i)T \)
good7 \( 1 + (-8.40 + 8.40i)T - 49iT^{2} \)
11 \( 1 + (-15.5 - 11.3i)T + (37.3 + 115. i)T^{2} \)
13 \( 1 + (1.80 - 11.3i)T + (-160. - 52.2i)T^{2} \)
17 \( 1 + (5.37 + 10.5i)T + (-169. + 233. i)T^{2} \)
19 \( 1 + (-1.22 + 3.78i)T + (-292. - 212. i)T^{2} \)
23 \( 1 + (-2.57 + 0.408i)T + (503. - 163. i)T^{2} \)
29 \( 1 + (-15.6 - 48.3i)T + (-680. + 494. i)T^{2} \)
31 \( 1 + (8.55 + 2.77i)T + (777. + 564. i)T^{2} \)
37 \( 1 + (-3.43 - 0.544i)T + (1.30e3 + 423. i)T^{2} \)
41 \( 1 + (31.5 + 43.4i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 + (27.2 + 27.2i)T + 1.84e3iT^{2} \)
47 \( 1 + (-9.52 + 18.7i)T + (-1.29e3 - 1.78e3i)T^{2} \)
53 \( 1 + (19.1 - 37.5i)T + (-1.65e3 - 2.27e3i)T^{2} \)
59 \( 1 + (20.4 + 28.1i)T + (-1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (75.2 + 54.6i)T + (1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 + (8.79 + 17.2i)T + (-2.63e3 + 3.63e3i)T^{2} \)
71 \( 1 + (-24.2 - 74.5i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (43.8 - 6.94i)T + (5.06e3 - 1.64e3i)T^{2} \)
79 \( 1 + (8.03 + 24.7i)T + (-5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (29.6 + 58.1i)T + (-4.04e3 + 5.57e3i)T^{2} \)
89 \( 1 + (70.0 + 50.8i)T + (2.44e3 + 7.53e3i)T^{2} \)
97 \( 1 + (21.1 - 41.5i)T + (-5.53e3 - 7.61e3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.42916248435467850777133600342, −10.84770777902710384271201657841, −10.00895953513513589470476619548, −8.876362261796682559507597969980, −7.34789811817369990066182094350, −6.85909011540081232614930171234, −4.82007060978332839303427809842, −4.32441846930543573996837214675, −3.45813664288578053223749183808, −1.79184165869897746945600495946, 1.51829310757626286297788606327, 3.05039469689289301064831467788, 4.31370988471546389115756886244, 5.58848132597976317123161018438, 6.38514099516026430636737021463, 7.87320136213196717292512687115, 8.277280583168731733473511606477, 8.998079271530911205164606785713, 11.31264061640036391631243366955, 11.74468344238562634348468465417

Graph of the $Z$-function along the critical line